Internal Rate of Return

Break-even rate of return

• When calculating an NPV, we usually also calculate a break-even rate of return, called the internal rate of return (IRR).
• For projects with negative cash flows early and positive cash flows late:
  • If IRR > required return, then NPV>0.
  • If IRR < required return, then NPV<0.
• If cash flows are reversed (positive early and negative late), then we are borrowing instead of investing, and we like low rates instead of high.

Interpretation for usual projects

• Usual = “negative early and positive late”
• Suppose we invested the negative cash flows in stocks or other assets rather than the project.
• We want to pull out the positive cash flows later from the stock account.
• What rate of return would we have to earn on the stocks for this to be feasible?
• Answer is IRR.

Usefulness of the IRR

• If IRR is very high or very low, then the required return doesn’t matter much.
  • For usual projects, very high \(\Rightarrow\) “take.”
  • For usual projects, very low \(\Rightarrow\) “reject”.
• When there is limited capital, we can sometimes calculate the best portfolio of projects by starting with the highest IRR projects and working down the list.

Example

• Cash flows = [-100, 30, 30, 30, 30]

IRR is 7.71%

Calculating IRR

• NPV is a polynomial in \(1/(1+r)\):

numpy financial or excel

import numpy_financial as npf

cash_flows = [-100, 30, 30, 30, 30]
irr = npf.irr(cash_flows)